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Kaldor-Kalecki stochastic model of business cycles 1

GABRIELA MIRCEA, 2MIHAELA NEAMŢU, 3LAURA CISMAŞ, 4DUMITRU OPRIŞ 1,2 Department of Economic Informatics and Statistics 3 Department of Economy 4 Department of Mathematics 1,2,3,4 West University of Timişoara 1,2,3 Pestalozzi Street, nr. 16A, 300115, Timişoara, Romania 4 Bd. V. Parvan, nr. 4, 300223, Timişoara, Romania 1 [email protected], [email protected], 3 [email protected], [email protected]

Abstract: - This paper is concerned with a stochastic delayed Kaldor-Kalecki nonlinear business cycle model of the income. It will take into consideration the investment demand in the form suggested by Rodano. We will analyze the deterministic and stochastic Kaldor-Kalecki models. The dynamics of the mean values and the square mean values of the model’s variables are set. Numerical examples are given in the end, to illustrate our theoretical results. Key-Words: Business cycle model, deterministic model with delay, stochastic delay system, Kaldor-Kalecki model stochastic system is presented and the locally asymptotic stability is analyzed by the variables’ mean and the square mean. Numerical simulations are carried out in Section 5. Finally, concluding remarks are given in Section 6.

1 Introduction The model proposed by Kaldor [3] is one of earliest and simplest nonlinear models of business cycles. This model cannot be considered as a satisfying description of actual economies. Nevertheless, it continues to generate a considerable amount of economic, pedagogical and methodological interest, both from the point of view of the economist and of the applied dynamicist. Kalecki introduced the idea that there is a time delay for investment before a business decision. Krawiek and Sydlowski [2] incorporated Kalecki’s idea into Kaldor’s model and proposing the KaldorKalecki model of business cycles. In recent literature, it has been proved that information delay makes dynamic economic models unstable. In situations where delay is important, models with stochastic perturbation are framed by stochastic differential delay equations. In this paper, we will investigate the effects of the random perturbation for Kaldor-Kalecki model analyzing the steady state of the model with stochastic perturbation. The reminder of the paper develops as follows. In Section 2, we describe a deterministic and stochastic Kaldor-Kalecki model using the investment demand proposed in Rodano [1]. In Section 3, we analyze the deterministic KaldorKalecki model, setting the conditions for the existence of the delay parameter value for which the model displays a Hopf bifurcation. In Section 4, the

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2 Deterministic and stochastic models of a Kaldor Kalecki business cycle with delay In the last decade, the study of delayed differential equations that arose in business cycles has received much attention. The first model of business cycles can be traced back to Kaldor [3], who used a system of ordinary differential equations to study business cycles in 1940 by proposing nonlinear investment and saving functions so that the system may have cyclic behaviors or limit cycles, which are important from the point of view of economics. Kalecki [2] introduced the idea that there is a time delay for investment before a business decision. Krawiec and Szydlowski [2] incorporated the idea of Kalecki into the model of Kaldor by proposing the following Kaldor-Kalecki model of business cycles: dY (t ) = α ( I (Y (t ), K (t )) − S (Y (t ), K (t ))) dt (1) dK (t ) = I (Y (t − τ ), K (t )) − β K (t ) dt

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dY (t ) δu = α (δu + r ( − K (t )) + f (Y (t ) − u) − δY (t )) β dt (5) δu dK (t ) = δu + r( − K (t − τ )) + f (Y (t − τ ) − u) − βK (t ) β dt

where Y is the gross product, K is the capital stock, α > 0 is the adjustment coefficient in the goods market, β ∈ (0,1) is the depreciation rate of capital stock, I (Y , K ) and S (Y , K ) are investment and saving functions, and τ ≥ 0 is a time lag representing delay for investment due to the past investment decision. Considering that past investment decisions also influence the change in the capital stock, in [10] we extended the model by imposing delays in both the gross product and capital stock. Thus, adding of same delay to the capital stock K in the investment function I (Y , K ) in the second equation of system (1), the following Kaldor-Kalecki model business cycles is obtained: dY (t ) = α ( I (Y (t ), K (t )) − S (Y (t ), K (t ))) dt (2) dK (t ) = I (Y (t − τ ), K (t − τ )) − βK (t ) dt

System (5) with the initial conditions: Y (θ ) = φ1 (θ ), K (θ ) = φ2 (θ ), θ ∈ [−τ ,0]

and φ1 , φ2 : [−τ ,0] → R of C1 class functions, represent a system of differential equations with delay. System (5) has the equilibrium given by δu Y0 = u , K 0 = .

β

Let (Ω, F , P) be the given probability space and w(t ) ∈ R be a scalar Wiener process on Ω , having independent stationary Gaussian increments with w(0) = 0, E ( w(t ) − w( s )) = 0 and where is the E ( w(t ) w( s )) = min(t , s ) , E mathematical expectation. The sample trajectories of w(t ) are continuous, nowhere differentiable and have infinite variations on any finite time interval [4]. For dynamical system (5), we are interested in knowing the effect of the noise perturbation on the equilibrium point (Y0 , K 0 ) . The stochastic disturbance model of system (5) is given by a system of stochastic differential equations with delay in the following way: δu dY (t ) = α (δu + r ( − K (t )) + f (Y (t ) − u) − δY (t ))dt − β − σ1 (Y (t ) − Y0 )dw(t ) (7) δu dK (t ) = δu + r ( − K (t − τ )) + f (Y (t ) − u)dt − β − σ 2 (K (t ) − K0 )dw(t ) where σ 1 > 0, σ 2 > 0. The solution of (7) is a stochastic process denoted by Y (t ) = Y (t , ω ), K (t ) = K (t , ω ), ω ∈ Ω . From the Chebyshev inequality, the possible range of Y , K at a time t is “almost” determined by its mean and variance at time t . So, the first and second moments are important for investigating the solution behavior.

As usual in a Keynesian framework, savings are assumed to be proportional to the current level of income: (3) S (Y , K ) = δY , where coefficient δ , δ ∈ (0,1) represents the propensity to save. As usual, the investment demand is assumed to be an increasing and sigmoid-shaped function of the income. Without loss of generality, in the following we shall consider the form proposed in Rodano [1]:  δu  (4) I (Y , K ) = δu + r  − K  + f (Y − u ) β  δu where is the “normal” level of capital stock. In

β

(4), two short-run investment components are considered: the first one is proportional to the difference between normal capital stock and current stock, according to a coefficient r (r > 0) , usually explained by the presence of adjustment costs; the second one is an increasing, but not linear, function of the difference between current income and its normal level. This second component of the short-run investment function is a convenient specification of the sigmoid-shaped relationship between investment and income proposed by Kaldor. We note that this analytic specification does not compromise the generality of the results. From (2) with (3) and (4) we obtain the following system:

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(6)

3 The analysis of Kaldor-Kalecki deterministic model The equilibrium point of system (5) is the solution of the following system:

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 δu

 − K  + f (Y − u ) − δY = 0 β  (8)  δu  δu + r  − K  + f (Y − u ) − βK = 0 β  From (8) we obtain the equilibrium point δu Y0 = u , K 0 = . By carrying out the translation

ω2 +αβ(ρ1 −δ ) = αδr cosθ + ωr sinθ (α(ρ1 −δ ) − β)ω = ωr cosθ −αδr sinθ

δu + r 

where θ = ωτ . From (14), by squaring each relation and their addition, we obtain equation (13). β (β + r ) From condition we get δ< αr δ +r β +r 0, a2 + c2 > 0, where a1 , a2 , c1 , c2 are given by (27). In this case the equation system that described the square mean is asymptotically stable.

}

Let R (t , s ) = E y (t ) y ( s ) be the covariance matrix of the process y (t ) so that R(t , t ) satisfies: dR (t , t ) = AR(t , t ) + R(t , t ) AT + BR(t − τ , t ) + dt (23) + R(t , t − τ ) BT + CR (t , t )C From (22) and A, B, C given by (17), we get: Proposition 4: 1. The differential system (23) is given by: dR11(t,t) = (2α(ρ1 − δ ) + σ12 )R11(t,t) − 2αrR12(t,t) dt dR22(t,t) = (−2β + σ 22 )R22(t,t) − 2rR22(t,t − τ ) + dt + 2ρ1R12(t − τ ,t)

Proposition 7: If τ ≠ 0 and the equation: (28) 16ω 4 + (16a12 − 8c2 − c12 ) + a22 − c22 = 0 has a positive root ω2 , then for τ ∈ (0,τ 2 ) the characteristic equation h(λ ,τ ) = 0 has roots with

negative real parts. Therefore, the square mean values are asymptotically stable and

(24)

dR12(t,t) = −αrR22(t,t) + α(ρ1 − δ − β − σ1σ 2 )R12(t, t) + dt + ρ1R11(t, t −τ ) − rR12(t,t − τ )

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(27)

c2 = 4αδr − 2rσ 12

= E{Ay(t) y (t) + y(t) y (t) A + By(t −τ ) y (t) + (22) T

where

setting the condition that the system we obtain should accept nontrivial solution, we get h (λ , τ ) = 0 . The stability of the second moment is done by analyzing the roots of the characteristic equation h (λ ,τ ) = 0 .

The mean of the solution for (16) behaves precisely like the solution of the unperturbed deterministic equation (10). The proof results from taking into account the mathematical expectation of both sides of (16) as well as the properties of the Ito calculus. To examine the stability of the second moment of y (t ) for linear stochastic differential delay equation (16) we use Ito’s rule to obtain the stochastic differential

{

i, j = 1,2.

K ij are constants. Replacing Rij (t , s ) in (23) and

Proposition 3: The moment of the solution of (6) is given by:

dE ( y (t )) = AE ( y (t )) + BE ( y (t − τ )) . dt

(25)

+ 2αrρ1(4λ + β −σ12 −σ22 − 2α(ρ1 −δ ) + 2re−λτ )

τ2 =

89

1

ω2

arctg

(4a1c2 − a2 c1 + 4c1ω2 )ω2 . 4ω2 (c2 − a1c1 ) − a2 c2

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Figures Fig.5, Fig.6 represent the orbits of the dispersions’ (t , D ( y1 (t )) and (t , D ( y2 (t )) :

For τ = τ 2 the system (24) has a limit cycle. The proof is similar as the one in the case of Proposition 1.

4 Numerical simulations For the numerical simulation, the following values were taken into consideration: α = 0.8 , β = 0.2 , δ = 0.3 , r = 2 , u = 3 , and the function 0.3 . The value of τ 0 given by f ( x) = −4x (1 + e ) − 0.5 (15’) is τ 0 = 0.77 . Therefore, for τ ∈ (0,τ 0 ) the differential system (5) is asymptotically stable. At the same

Fig.5 The orbit (t , D ( y1 (t ))

time, the mean values of system (21) are asymptotically stable. If σ 1 = σ 2 = 0.283239697 and τ ∈ (0,τ 2 ) where τ 2 = 0.7585959136 the system of the square mean’s values is asymptotically stable. Because τ 2 < τ1 for τ ∈ (0,τ 2 ) , the variances

Fig.6 The orbit (t , D ( y2 (t ))

Figures Fig.7, Fig.8 represent the orbits of the mean values (t , y1 (t , ω )) and (t , y2 (t , ω )) :

given by: D ( y1 (t )) = E ( y1 (t )) 2 − E ( y1 (t ) 2 ) D ( y2 (t )) = E ( y2 (t )) 2 − E ( y2 (t ) 2 ) are asymptotically stable.

Fig.7 The orbit (t , y1 (t , ω ))

Figures Fig.1, Fig.2 represent the orbits of the mean values (t , E ( y1 (t )) and (t , E ( y2 (t )) :

A similar analysis can be carried out for the following functions: f1 ( x) = 0.2 tan( x ) , f 2 ( x ) = 0.2 arctan h( x) , f 3 ( x) = 0.2 sin( x) , f 4 ( x) =

The analysis of a Kaldor-Kalecki business cycle model in this paper allowed us to obtain some new dynamic scenarios which may be interesting both for the applied dynamicist and the economist. The paper has analyzed the Kaldor-Kalecki model and the steady state of model with stochastic perturbation. For the stochastic model, we have analyzed the square mean and the depression of the model’s variables. We have determined the values of the delay for which Kaldor-Kalecki system is asymptotically

square mean’s values (t , R( y1 (t ) 2 ) and (t , R( y2 (t ) 2 ) :

(t , R( y1 (t ) )

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stable and for which the system displays a limit cycle. We have determined the values of τ for

Fig.4 The orbit 2

ex . 1+ ex

5 Conclusion

Fig.1 The orbit Fig.2 The orbit (t , E ( y2 (t )) (t , E ( y1 (t )) Figures Fig.3, Fig.4 represent the orbits of the

Fig.3 The orbit

Fig.8 The orbit (t , y 2 (t , ω ))

which the square mean’s values and the variances are stable.

(t , R( y2 (t ) 2 )

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business cycles with delay, Discrete dynamics in Nature and Society, 2009, doi:10.1155/2009/ 923809, (in press).

Acknowledgement: The research was done under the Grant with the title “The qualitative analysis and numerical simulation for some economic models which contain evasion and corruption”, CNCSIS-UEFISCU Romania (grant No. 1085/2008). References: [1] Bischi G. I., Dieci R., Rodano G., Saltari E. Multiple attractors and global bifurcations in a Kaldor-type business cycle model, Journal Evolutionary Economic No. 11 (2001), pp. 527554; [2] Kaddar A., Talibi Aluoui H. - Global existence of periodic solutions in a delayed Kaldor-Kalecki model (in press) [3] Kaldor N. - A model of the trade cycle, Economic Journal No. 50, pp 78-92; [4] Kloeden P.E., Platen E. - Numerical Solution of Stochastic Differential Equations, Springer Verlag, Berlin, 1995; [5] Mircea G., Neamţu M., Opriş D. - Hopf bifurcations for dynamical systems with time delay and application (in Romanian), Mirton Publishing House, Timişoara, 2004; [6] Mircea G., Nemţu M., Ciurdariu A.L., Opriş D. Numerical simulation for Dynamic Stochastic Models of Internet Networks, WSEAS Transactions on Mathematics, Issue 2, Volume 8, February 2009, ISSN: 1109-2769, pp. 679-688. [7] Mircea G., Pirtea M., Neamţu M., Opriş D. - The stochastic monetary model with delay, Proceedings of 2nd WSEAS World Multiconference-Applied Economics, Bussiness and Development, Sousse, Tunisia, 2010 (in press); [8] Neamţu M., Mircea G., Opriş D. - The Study of Some Discrete IS-LM Models with Tax Revenues and Time Delay, WSEAS Transactions on Mathematics, Issue 2, Volume 8, February 2009, ISSN: 1109-2769, pp. 51-62. [9] Neamţu M., Mircea G., Opriş D. - A Discrete ISLM Model with Tax Revenues, Proceedings of the 10th WSEAS International Conference on Mathematics and Computers in Business and Economics (MCBE’09) - “Recent Advanced in Mathematics and Computers in Business and Economics”, Prague, 2009, pp.171-176. [10] Neamţu M., Pirtea M., Mircea G., Opriş D. Stochastic delayed dynamics heterogeneous competition with product differentiation, Proceedings of 2nd World Multiconference Applied Economics, Bussiness and Development, Sousse, Tunisia, 2010 (in press); [11] Wu P.X. - Simple zero and double zero singularities of a Kaldor-Kalecki model of

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